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Keywords:
directed complete poset; Scott topology; dcpo-completion; partial dcpo; C-space; lattice of continuous functions; lower semicontinuous functions; injective hull
directed complete poset; Scott topology; dcpo-completion; partial dcpo; C-space; lattice of continuous functions; lower semicontinuous functions; injective hull
Summary:
We introduce partial dcpo’s and show their some applications. A partial dcpo is a poset associated with a designated collection of directed subsets. We prove that (i) the dcpo-completion of every partial dcpo exists; (ii) for certain spaces $X$, the corresponding partial dcpo’s of continuous real valued functions on $X$ are continuous partial dcpos; (iii) if a space $X$ is Hausdorff compact, the lattice of all S-lower semicontinuous functions on $X$ is the dcpo-completion of that of continuous real valued functions on the space; (iv) a topological space has an injective hull iff it is homeomorphic to the pre-Scott space of a continuous partial dcpo whose way-below relation satisfies the interpolation property.
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